JavaEE is a set of libraries and tools designed for creating Java servers.
Spring is a framework that runs on top of JavaEE.
I’ve used Spring, but IMHO it’s overkill for everything I’ve needed to do. I could see it being useful for a team of developers working on a full product, but I haven’t really needed it for my purposes. (Static Void Games was written in Spring, and it probably caused more headaches than it solved.)
If you want to explore frameworks, I’d personally start with Struts:
OpenJDK has been around for a while. From what I understand, from a developer’s perspective it’s not really much different from regular Java. The difference is in the underlying implementation. So it’s not like you have to learn one over the other. The difference is more important when you have to worry about licensing and deployment. (I’m not an expert so I could be getting that wrong, but that’s my understanding.)